One-point extensions of Markov processes by darning

This paper is a continuation of the works by Fukushima-Tanaka (Ann Inst Henri Poincare Probab Stat 41: 419-459, 2005) and Chen-Fukushima-Ying (Stochastic Analysis and Application, p.153-196. The Abel Symposium, Springer, Heidelberg) on the study of one-point extendability of a pair of standard Markov processes in weak duality. In this paper, general conditions to ensure such an extension are given. In the symmetric case, characterizations of the one-point extensions are given in terms of their Dirichlet forms and in terms of their L-2-infinitesimal generators. In particular, a generalized notion of flux is introduced and is used to characterize functions in the domain of the L-2-infinitesimal generator of the extended process. An important role in our investigation is played by the alpha-order approaching probability u(alpha).